arXiv:1302.4091v2 [math.DS] 29 Jun 2013w3.impa.br/~cmateus/files/AMY1.pdf · 2 A. AVILA, C. MATHEUS AND J.-C. YOCCOZ relationship with the moduli spaces H gof normalized (unit area)

SL(2,R)-INVARIANT PROBABILITY MEASURES ON THE MODULI SPACESOF TRANSLATION SURFACES ARE REGULAR

ARTUR AVILA, CARLOS MATHEUS AND JEAN-CHRISTOPHE YOCCOZ

ABSTRACT. In the moduli space Hg of normalized translation surfaces of genus g, con-sider, for a small parameter ρ > 0, those translation surfaces which have two non-parallelsaddle-connections of length6 ρ. We prove that this subset of Hg has measure o(ρ2) w.r.t.any probability measure on Hg which is invariant under the natural action of SL(2,R).This implies that any such probability measure is regular, a property which is importantin relation with the recent fundamental work of Eskin-Kontsevich-Zorich on the Lyapunovexponents of the KZ-cocycle.

CONTENTS

1. Introduction 11.1. Regular SL(2,R)-invariant probability measures on Hg 11.2. The result 31.3. Reduction to a particular case 41.4. Scheme of the proof 51.5. Notations 61.6. Two basic facts 6Acknowledgements 62. Conditional measures 62.1. The general setting 62.2. A special setting 73. Preliminaries on SL(2,R) 93.1. The decomposition gtRθnu 93.2. Euclidean norms along a SL(2,R)-orbit 93.3. On the action of the diagonal subgroup 104. Construction of a measure related to m 125. Measure of the slice {ρ > sys(M) > ρ exp(−τ)} 145.1. The regular part of the slice 145.2. Estimate for the singular part of the slice 156. Proof of Theorem 1.3 17References 20

1. INTRODUCTION

1.1. Regular SL(2,R)-invariant probability measures on Hg . The study of intervalexchange transformations and translation flows on translation surfaces (such as billiards inrational polygons) was substantially advanced in the last 30 years because of its intimate

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2 A. AVILA, C. MATHEUS AND J.-C. YOCCOZ

relationship with the moduli spaces Hg of normalized (unit area) Abelian differentials oncompact Riemann surfaces of genus g > 1. In fact, the moduli spaces of normalizedAbelian differentials admit a natural action of SL(2,R) and this action works as a renor-malization dynamics for interval exchange transformations and translation flows. In par-ticular, this fundamental observation was successfully applied by various authors to deriveseveral remarkable results:

• in the seminal works of H. Masur [12] and W. Veech [18] in the early 80’s, therecurrence properties of the orbits of the diagonal subgroup gt = diag(et, e−t)of SL(2,R) on Hg were a key tool in the solution of Keane’s conjecture on theunique ergodicity of typical interval exchange transformations;

• after the works of A. Zorich [22], [23] in the late 90’s and G. Forni [8] in 2001,we know that the Lyapunov exponents of the action of the diagonal subgroupgt = diag(et, e−t) of SL(2,R) on Hg drive the deviations of ergodic averagesof typical interval exchange transformations and translation flows;

• more recently, V. Delecroix, P. Hubert and S. Lelièvre [3] confirmed a conjectureof the physicists J. Hardy and J. Weber on the abnormal rate of diffusion of typi-cal trajectories in typical realizations of the Ehrenfest wind-tree model of Lorenzgases by relating this question to certain Lyapunov exponents of the diagonal sub-group gt = diag(et, e−t) of SL(2,R) on H5.

It is worth to point out that the applications in the last two items above involved Lya-punov exponents of the diagonal subgroup gt = diag(et, e−t) of SL(2,R) on the modulispaces of normalized Abelian differentials, and this partly explains the recent literaturededicated to this subject (e.g., [1], [2], [4], [9] and [15]).

In this direction, the most striking recent result is arguably the formula of A. Eskin,M. Kontsevich and A. Zorich [5] relating sums of Lyapunov exponents to certain geomet-rical quantities called Siegel-Veech constants. Very roughly speaking, the derivation ofthis formula can be described as follows (see also the excellent Bourbaki seminar [10] byGrivaux-Hubert on this subject). Given m an ergodic SL(2,R)-invariant probability mea-sure on Hg , a remarkable formula of M. Kontsevich [11] and G. Forni [8] allows to expressthe sum of the top g Lyapunov exponents of m in terms of the integral of the curvature ofthe determinant line bundle of the Hodge bundle over the support of m. The formula ofKontsevich and Forni suffices for some particular examples of m, but, in general, its rangeof applicability is limited because it is not easy to compute the integral of the curvature ofthe determinant line bundle of the Hodge bundle. For this reason, A. Eskin, M. Kontse-vich and A. Zorich use an analytic version of the Riemann-Roch-Hirzebruch-Grothendiecktheorem to express the integral of the curvature of the determinant of the Hodge bundle

as the sum of a simple combinatorial term 112σ∑i=1

ki(ki+2)ki+1

depending only on the orders

k1, . . . , kσ of the zeroes of the Abelian differentials in the support of m and a certain in-tegral expression I (that we will not define explicitly here) depending on the flat geometryof the Abelian differentials in the support ofm. At this point, A. Eskin, M. Kontsevich andA. Zorich complete the derivation of their formula with an integration by parts argumentin order to relate this integral expression I mentioned above to Siegel-Veech constants as-sociated to the problem of counting maximal (flat) cylinders of the Abelian differentials inthe support of m.

Concerning the hypothesis for the validity of the formula of A. Eskin, M. Kontsevichand A. Zorich, as the authors point out in their article [5], most of their arguments useonly the SL(2,R)-invariance of the ergodic probability measure m. Indeed, the sole place

REGULARITY OF SL(2,R)-INVARIANT MEASURES ON MODULI SPACES 3

where they need an extra assumption on m is precisely in Section 9 of [5] for the justi-fication of the integration by parts argument mentioned above (relating a certain integralexpression I to Siegel-Veech constants).

Concretely, this extra assumption is called regularity in [5] and it is defined as follows(see Subsections 1.5 and 1.6 of [5]). Given a normalized Abelian differential ω on a Rie-mann surface M of genus g > 1, we think of (M,ω) ∈ Hg as a translation surface,that is, we consider the translation (flat) structure induced by the atlas consisting of localcharts obtained by taking local primitives of ω outside its zeroes. Recall that a (maxi-mal flat) cylinder C of (M,ω) is a maximal collection of closed parallel regular geodesicsof the translation surface (M,ω). The modulus mod(C) of a cylinder C is the quotientmod(C) = h(C)/w(C) of the height h(C) of C (flat distance across C) by the lengthw(C) of the waist curve of C. In this language, we say that an ergodic SL(2,R)-invariantprobability measure m on Hg is regular if there exists a constant K > 0 such that

limρ→0

m(Hg(K, ρ))

ρ2= 0, i.e., m(Hg(K, ρ)) = o(ρ2),

where Hg(K, ρ) is the set consisting of Abelian differentials (M,ω) ∈ Hg possessing twonon-parallel cylinders C1 and C2 with mod(Ci) > K and w(Ci) 6 ρ for i = 1, 2.

Remark 1.1. A saddle-connection on a translation surface (M,ω) ∈ C is a geodesicsegment joining zeroes of ω without zeroes of ω in its interior. The systole of M , denotedby sys(M), is the length of the shortest saddle-connection on M .

Each boundary component of a cylinder consists of unions of saddle-connections, sothat sys(M) 6 w(C) for any cylinder C of (M,ω).

It follows from the Siegel-Veech formula (cf. [19], Theorem 2.2 in [6], and Lemma 9.1in [5]) that we have, for small ρ > 0

m({sys(M) 6 ρ}) = O(ρ2).A fortiori, those surfaces (M,ω) ∈ Hg with a cylinder C satisfying w(C) 6 ρ form a setof m-measure O(ρ2).

As it is mentioned by A. Eskin, M. Kontsevich and A. Zorich right after Definition 1in [5], all known examples of ergodic SL(2,R)-invariant probabilities m are regular: forinstance, the regularity of Masur-Veech (canonical) measures was shown in Theorem 10.3in [14] and Lemma 7.1 in [7] (see also [16] for recent related results). In particular, this ledA. Eskin, M. Kontsevich and A. Zorich to conjecture (also right after Definition 1 in [5])that all ergodic SL(2,R)-invariant probabilities m on moduli spaces Hg of normalizedAbelian differentials are regular.

1.2. The result. In this paper, we confirm this conjecture by showing the following slightlystronger result. For ρ > 0, we denote by Hg,(2)(ρ) the set of M ∈ Hg which have at leasttwo non-parallel saddle-connections of length 6 ρ.

Theorem 1.2. Let m be a SL(2,R)- invariant probability measure on Hg . For smallρ > 0, the measure of Hg,(2)(ρ) satisfies

m(Hg,(2)(ρ)) = o(ρ2).

The moduli space Hg is the finite disjoint union of its strata H(1)(k1, . . . , kσ). Here,

(k1, . . . , kσ) runs amongst non-increasing sequences of positive integers withσ∑i=1

ki =


2g − 2, and H(1)(k1, . . . , kσ) consists of unit area translation surfaces (M,ω) ∈ Hg suchthat the 1-form ω has σ zeroes of respective order k1, . . . , kσ .

Every stratum is invariant under the action of SL(2,R). Therefore, if m is a SL(2,R)-invariant probability measure on Hg as in the theorem, we may consider its restrictionto each stratum of Hg (those who have positive measure) and deal separately with theserestrictions.

It means that in the proof of the theorem, we may and will assume that m is supportedby some stratum H(1)(k1, . . . , kσ) of Hg . Actually, we may even allow for some of theintegers ki to be equal to zero, corresponding to marked points on the translation surfacewhich are not zeroes of the 1-form ω.

A slightly more general form of our theorem is thus:

Theorem 1.3. Let g > 0, σ > 0 and let (k1, . . . , kσ) be a non-increasing sequence

of non-negative integers withσ∑i=1

ki = 2g − 2. Let H(1) = H(1)(k1, . . . , kσ) be the

corresponding moduli space of unit area translation surfaces. For ρ > 0, let H(1)(2)(ρ) be

the subset consisting of translation surfaces in H(1) which have at least two non-parallelsaddle-connections of length 6 ρ.

For any SL(2,R)-invariant probability measure m on H(1), one has

m(H(1)(2)(ρ)) = o(ρ

2).

1.3. Reduction to a particular case. The moduli space H(1)(k1, . . . , kσ) is only an orb-ifold, not a manifold, because some translation surfaces have non trivial automorphisms.We explain here how to bypass this (small) difficulty.

Denote by H = H(k1, . . . , kσ) the moduli space of translation surfaces with combina-torial data (k1, . . . , kσ) of any positive area, so that H(1) is the quotient of H by the actionof homotheties of positive ratio.

The moduli space H(k1, . . . , kσ) is the quotient of the corresponding Teichmüller spaceQ(k1, . . . , kσ) by the action of the mapping class group MCG(g, σ). Here, a subset Σ ={A1, . . . , Aσ} of a (compact, connected, oriented) surface M0 of genus g is given. TheTeichmüller space Q(k1, . . . , kσ) is the set of of structures of translation surfaces on M0having a zero of order ki at Ai, up to homeomorphisms of M0 which are isotopic to theidentity rel.Σ. The mapping class group MCG(g, σ) is the group of isotopy classes rel.Σof homeomorphisms of M0 preserving Σ.

The Teichmüller space Q(k1, . . . , kσ) is a complex manifold of dimension 2g + σ − 1with a natural affine structure given by the period map

Θ : Q(k1, . . . , kσ) −→ Hom (H1(M0,Σ,Z),C) = H1(M0,Σ,C)

ω 7−→ (γ 7−→∫γ

ω)

which is a local homeomorphism.

The mapping class group acts properly discontinuously on Teichmüller space. However,some points in Teichmüller space may have non trivial stabilizer in MCG(g, σ), leadingto the orbifold structure for the quotient space. The stabilizer of a structure of translationsurface ω on (M0,Σ) is nothing but the finite group Aut(M0, ω) of automorphisms of thisstructure. The automorphism group acts freely on the set of vertical upwards separatrices


of ω. Therefore its order is bounded by the number of such separatrices∑i(ki + 1) =

2g − 2 + σ.

Let ω0 be a structure of translation surface on (M0,Σ); let G be its automorphismgroup, viewed as a finite subgroup of MCG(g, σ). For ω close to ω0 in Teichmüllerspace, the automorphism group Aut(M0, ω) is contained in G, because the action of themapping class group is properly discontinuous. Moreover, for any g ∈ G, one has g ∈Aut(M0, ω) iff Θ(ω) is a fixed point of g (for the natural action of the mapping classgroup on H1(M0,Σ,C)). Thus, in a neighborhood of ω0, those ω which have the sameautomorphism group than ω0 form an affine submanifold of Teichmüller space, and theimage of this subset in the moduli space H is a manifold. The intersection of this imagewith H(1) is also a manifold.

For i > 1, denote by Ci the set of points of H(1)(k1, . . . , kσ) for which the automor-phism group (defined up to conjugacy in the mapping class group) has order i. It followsfrom the previous discussion that Ci is a manifold for all i, and is empty for i > 2g−2+σ.Moreover, each Ci is invariant under the action of SL(2,R).

Let m be a SL(2,R)-invariant probability measure supported on H(1). To prove theproperty of Theorem 1.3, it is sufficient to prove it for the restriction of m to each of theCi (those which have positive measure). Therefore we may and will assume below that mis supported on some Ci.

1.4. Scheme of the proof. The basic idea of the proof of Theorem 1.3 is the follow-ing. We consider, in a level set {sys(M) = ρ0} of the systole function on Ci, the subsetof translation surfaces whose shortest saddle-connections are all parallel. We will usethe SL(2,R)-action to move relatively large parts of this subset to further deep sublevels{sys(M) 6 ρ := ρ0 exp(−T )} for T � 1. The translation surfaces obtained in this wayhave the property that all saddle-connections not parallel to a minimizing one are muchlonger than the systole. A computation in appropriate pieces of SL(2,R)-orbits, on whichthe invariant probability desintegrate as Haar measure, shows that most surfaces in the sub-level {sys(M) 6 ρ} are obtained in this way. Thus we can conclude that the complementin Ci, which contains the intersection Hg,(2)(ρ) ∩ Ci has measure o(ρ2).

The article is organized as follows. In Section 2 , we review some material on Rokhlin’sdesintegration theorem ([17], see also [21]), which allows to consider separately each orbitof the action of SL(2,R). The statement that we aim at (Proposition 2.6) is well-known tospecialists, but we were not able to find a proper reference in the literature.

In Section 3, we discuss a couple of elementary facts about SL(2,R) and its action onR2 (see Proposition 3.1, Proposition 3.3 and Proposition 3.4) lying at the heart of our “orbitby orbit” estimates.

In Section 4, we construct from the invariant probability m a related measure m0 whichhas finite total mass and is supported on the subset X∗0 of the level set {sys(M) = ρ0}formed by surfaces on which all minimizing saddle-connections are vertical. This mea-sure m0 enters in the formula (Corollary 4.2) for the m-measure of certain subsets of thesublevels {sys(M) 6 ρ0 exp(−T )}.

In Section 5, we show that the measure of slices of the form {ρ0 > sys(M) > ρ0 exp(−τ)}when τ is small is related to the total mass of m0 (Corollary 5.4). The slice is divided intoa regular part, obtained by pushing X∗0 into the slice through the action of SL(2,R), anda singular part , whose measure is much smaller (Proposition 5.3).


In Section 6, we bring together the results of the previous sections to present the proofof the theorem.

1.5. Notations. We will assume some familiarity with the basic features of Abelian dif-ferentials, translation surfaces and their moduli spaces: we refer the reader to the surveys[24] and [20] for gentle introductions to the subject. We will use the following notations:

• C denotes what was denoted by Ci above, namely the subset of a moduli spaceH(1)(k1, . . . , kσ) formed by points whose stabilizer in the mapping class grouphas a given order.

• m is a SL(2,R)-invariant probability measure supported on the manifold C.• (e1, e2) denotes the standard basis of R2 and || || is the usual Euclidean norm onR2.

• Rθ ∈ SO(2,R) is the rotation of angle θ.• gt is the diagonal matrix diag(et, e−t) in SL(2,R).

• nu is the lower triangular matrix(

1 0u 1

).

• Na,b is the upper triangular matrix(a b0 a−1

).

1.6. Two basic facts. We use several times the following lemma, which is an immediateconsequence of Fubini’s theorem.

Lemma 1.4. Let (X,B,m) be a probability space and let G be a locally compact groupacting measurably on X by measure-preserving automorphisms, and let ν be some Borelprobability measure on G. For any measurable subset B ⊂ X , one has

m(B) =

∫X

ν({g ∈ G, g.x ∈ B}) dm(x).

We will apply this taking for ν the normalized restriction of a Haar measure on G tosome compact subset.

We will use in the proof of Lemma 5.6 the following fundamental fact on moduli spaceof translation surfaces (see [6], Theorem 5.4 and also [13] for a much stronger result).

Lemma 1.5. For ρ > 0, R > 0, and any (M,ω) ∈ H(1)(k1, . . . , kσ) with sys(M) > ρ,the number of holonomy vectors of saddle-connections of (M,ω) of norm 6 R is boundedby a constant which depends only on ρ, R and k1, . . . , kσ .

Acknowledgements. The authors are thankful to Alex Eskin for several comments onsome preliminary versions of this paper. This work was supported by the ERC StartingGrant “Quasiperiodic”, by the Balzan project of Jacob Palis and by the French ANR grant“GeoDyM” (ANR-11-BS01-0004).

2. CONDITIONAL MEASURES

2.1. The general setting. We start by recalling the content of Rokhlin’s desintegrationtheorem ([17], see also [21]).

Definition 2.1. A probability space (X,B,m) is Lebesgue if either it is purely atomicor its continuous part is isomorphic mod.0 to [0, a] equipped with the standard Lebesguemeasure. Here a ∈ (0, 1] is the total mass of the continuous part of m.


Definition 2.2. A Polish space is a topological space homeomorphic to a separable com-plete metric space.

Any open or closed subset of a Polish space is Polish.Any Borel probability measure on a Polish space is Lebesgue. We generally omit in this

case the reference to the σ-algebra B, which is the σ-algebra generated by the Borel setsand the subsets of Borel sets of measure 0.

For a partition ζ of a set X , we denote by ζ(x) the element of ζ that contains a pointx ∈ X .

Definition 2.3. Let (X,B,m) be a Lebesgue probability space. A measurable partitionof X is a partition ζ of X which is the limit of a monotonous sequence (ζn)n>0 of finitepartitions by elements of B. This means that, for all x ∈ X, n > 0, one has

ζn(x) ∈ B, ζn+1(x) ⊂ ζn(x), ζ(x) = ∩n>0ζn(x).

As (X,B,m) is Lebesgue, the partition of X by points is measurable (mod.0).

Definition 2.4. Let ζ be a measurable partition of X . A system of conditional measuresfor (X,B,m, ζ) is a family (mx)x∈X of probability measures on (X,B) satisfying thefollowing properties:

• For any x ∈ X , one has mx(ζ(x)) = 1.• For any x, y ∈ X such that ζ(x) = ζ(y), one has mx = my .• For any B ∈ B, the function x 7→ mx(B) is measurable and one has

m(B) =

∫X

mx(B) dm(x).

The content of Rokhlin’s theorem is that such a system of conditional measures alwaysexists, and is essentially unique in the following sense: if (m′x)x∈X is another such system,then mx = m′x for m-a.a x.

2.2. A special setting. Let X be a Polish space, and let G be a Lie group. We will denoteby ν some given left invariant Haar measure onG, by d some given left invariant Riemann-ian distance on G, and by p1 the canonical projection from X × G onto X . We let G acton the left on X × G by g.(x, h) := (x, gh), i.e the product of the trivial action by thestandard action.

Let Z be a non empty open subset of X ×G, and let m be a Borel probability measureon Z. As an open subset of a Polish space, Z is a Polish space. Thus (Z,m) is a Lebesgueprobability space.

We will denote by m1 the Borel probability measure (p1)∗m on the open subset p1(Z)of X , and by Zx = {x} × Ux the fiber of Z over x. Here Ux is an open subset of G. Thepartition ζ of Z defined by ζ(x, g) = Zx is measurable.

Let (m(x,g))(x,g)∈Z be a system of conditional measures for (Z,m, ζ). From the secondproperty in the definition of conditional measures, the measure m(x,g) does not depend onthe second variable g. We will write mx instead of m(x,g). For each x ∈ p1(Z), mx maybe seen as a probability measure on Ux.

Definition 2.5. The Borel probability measure m on Z is invariant if, for any measurablesubset W ⊂ Z, any g ∈ G such that g.W ⊂ Z, one has m(g.W ) = m(W ).


Proposition 2.6. Assume that m is invariant. Then, for m1-a.e x, the non empty open setUx has finite Haar measure and we have

mx =1

ν(Ux)ν|Ux .

Proof. Choose a countable dense subset D ⊂ G and denote by B the set of closed balls(for the distance d) in G with center at a point of D and positive rational radius. The proofof the proposition is an easy consequence of the following elementary lemma, whose proofwill be given afterwards.

Lemma 2.7. Let U be a non empty open subset of G and let µ be a probability measureon U . Assume that, for any B ∈ B, g ∈ D such that both B and g.B are contained in U ,one has µ(g.B) = µ(B). Then U has finite Haar measure and µ = 1ν(U)ν|U .

In view of the conclusion of the lemma, the conclusion of the proposition will followif we know that, for m1-a.e x, the measure mx has the invariance property stated in thehypothesis of the lemma. Let B ∈ B, g ∈ D. The set X(B, g) formed of x ∈ X such thatboth B and g.B are contained in Ux is open in X . For any m1-measurable Y ⊂ X(B, g),one has, as m is invariant,∫

Y

mx(B) dm1(x) = m(Y ×B)

= m(Y × g.B)

=

∫Y

mx(g.B) dm1(x).

It follows that mx(B) = mx(g.B) for m1-a.a x ∈ X(B, g). Taking a countable in-tersection over the possible B, g gives the assumption of the lemma so the proof of theproposition is complete. �

Proof of lemma 2.7. We first show that µ is absolutely continuous w.r.t. the Haar measureν, i.e belongs to the Lebesgue class. Let B be any closed ball in B of small radius r > 0contained in U . There exists an integer M > Cr− dimG (where the constant C > 0depends on U but not on B) and elements g1, . . . , gM ∈ D such that the balls gi.B aredisjoint and contained in U . Then we have, from the assumption of the lemma

Mµ(B) = µ(

M⋃1

gi.B) 6 µ(U) = 1.

It follows that µ(B) 6 C−1rdimG for all B ∈ B contained in U . This implies that µbelongs to the Lebesgue class and that its density φ w.r.t. the Haar measure ν is bounded.By the Lebesgue density theorem, for ν-almost all x ∈ U , one has

φ(x) = limB3x

µ(B)

ν(B)

where the limit is taken over balls in B containing x with radii converging to 0. If x, x′ areany two points of U with this property, one can find a sequence Bi of balls in B with radiiconverging to 0 and a sequence gi ∈ D such that x ∈ Bi ⊂ U and x′ ∈ gi.Bi ⊂ U for alli. It follows from the invariance assumption on µ and the left invariance of Haar measurethat φ(x) = φ(x′). Thus the density φ is constant ν-almost everywhere in U and the proofof the lemma is complete. �


Remark 2.8. In the defining property of conditional measures∫Z

f(x, g) dm(x, g) =

∫p1(Z)

(∫Ux

f(x, g) dmx(g)

)dm1(x)

bothm1 andmx are probability measures. In the context of Proposition 2.6, one may alsowrite ∫

Z

f(x, g) dm(x, g) =

∫p1(Z)

(∫Ux

f(x, g) dν(g)

)(ν(Ux))

−1dm1(x).

However, the measure (ν(Ux))−1dm1(x) is not necessarily finite.

3. PRELIMINARIES ON SL(2,R)

3.1. The decomposition gtRθnu. Let W be the set of matrices(a bc d

)in SL(2,R)

such that d > 0 and |bd| < 12 .

Proposition 3.1. The map (t, θ, u) 7→ gtRθnu from R× (−π4 ,π4 )×R is a diffeomorphism

onto W . Using this map to take (t, θ, u) as coordinates on W , the restriction to W of (aconveniently scaled version of ) the Haar measure is equal to cos 2θ dt dθ du.

Proof. The vertical basis vector e2 is fixed by nu. For |θ| < π4 and t ∈ R, its image (b, d)under gtRθ satisfy d > 0 and |bd| < 12 .

Conversely, given (b, d) satisfying these conditions, there is a unique pair (t, θ) ∈ R ×(−π4 ,

π4 ), depending smoothly on (b, d), such that (b, d) = gtRθ.e2. The first assertion of

the proposition follows.

Write the restriction to W of the Haar measure on SL(2,R) as ψ(t, θ, u)dt dθ du, forsome positive smooth function ψ on R × (−π4 ,

π4 ) × R. As the Haar measure is left-

invariant and right-invariant, the function ψ does not depend on t and u, only on θ. Asmall calculation, using the left-invariance under Rθ, shows that ψ(θ) = cos 2θ. �

3.2. Euclidean norms along a SL(2,R)-orbit. The following lemma is elementary andwell-known.

Lemma 3.2. The map (θ, a, b) 7→ RθNa,b from R/2πZ × R>0 × R to SL(2,R) (corre-sponding to the Iwasawa decomposition) is a diffeomorphism. The measure dθ da db issent to a Haar measure by this diffeomorphism.

Proposition 3.3. Let v, v′ be vectors in R2 with v 6= ±v′. The Haar measure of the setE(v, v′, τ) consisting of the elements γ ∈ SL(2,R) such that

||γ|| 6 2, exp(−τ) 6 ||γ.v|| 6 exp τ, exp(−τ) 6 ||γ.v′|| 6 exp τ

is O(τ32 ) when τ is small; the implied constants are uniform when ||v ± v′|| is bounded

away from 0.

Proof. The set E(v, v′, τ) is empty (for small τ ) unless 13 6 ||v|| 6 3,13 6 ||v

′|| 6 3, sowe may assume that v, v′ are constrained by these inequalities. We may also assume thatv = (p, 0) with 13 6 p 6 3. We write v

′ = (q, r), and γ = RθNa,b as in the Lemma. Thefunction

N : γ 7→ (||γ.v||2, ||γ.v′||2)does not depend on θ; one has

N(a, b) = (p2a2, (qa+ rb)2 + r2a−2).


The Jacobian matrix of N is

2

(p2a 0

q(qa+ rb)− r2a−3 r(qa+ rb)

).

As a > 0, this matrix is invertible unless r(qa + rb) = 0, i.e γ.v, γ.v′ are collinear ororthogonal. We conclude as follows

• Assume first that v′ = ±(λv + w) with 0 < λ 6= 1, ||w|| < 1100 |λ − 1|. Thenγ.v′ = ±(λγ.v + γ.w) with ||γ.w|| < 2100 |λ − 1| when ||γ|| 6 2. As λ 6= 1, wecannot have at the same time

exp(−τ) 6 ||γ.v|| 6 exp τ, exp(−τ) 6 ||γ.v′|| 6 exp τif τ is small enough. The set E(v, v′, τ) is thus empty for small τ ; the implied

constant depends only on |λ− 1|, i.e on ||v ± v′||.• Assume next that v, v′ are orthogonal, i.e q = 0. Then N(a, b) = (p2a2, r2(b2 +a−2)). The condition exp(−τ) 6 ||γ.v|| 6 exp τ determines an interval for aof length O(τ). For each value of a in this interval, the condition exp(−τ) 6||γ.v′|| 6 exp τ determines an interval for b whose length is O(τ 12 ) ( and exactlyof order τ

12 in the worst case pr = 1). Thus, the Haar measure of E(v, v′, τ)

is O(τ32 ). Observe that this is still true if we relax the condition ||γ|| 6 2 to

||γ|| 6 6.• Assume that there exists γ0 ∈ SL(2,R) such that ||γ0|| 6 3 and γ0.v, γ0.v′

are orthogonal. Then the required estimate is a consequence of the previous caseafter translating by γ0, taking into account the observation concluding the previousdiscussion.

• Finally, assume that none of the above holds. Then the Jacobian matrix of Non ||γ|| 6 2 is everywhere invertible and the norm of its inverse is uniformlybounded when ||v ± v′|| is bounded away from 0. In this case, the Haar measureofE(v, v′, τ) isO(τ2), with a uniform implied constant when ||v±v′|| is boundedaway from 0.

The proposition is proved. �

3.3. On the action of the diagonal subgroup. LetRθ be some given rotation. For T > 0,consider the set

J(T, θ) := {t ∈ R, ||gtRθ.e2|| < exp(−T )}.

Proposition 3.4. The set J(T, θ) is empty iff | sin 2θ| > exp(−2T ). When | sin 2θ| <exp(−2T ), writing sin 2θ = e−2T sinω with cosω > 0, the set J(T, θ) is an open intervalof length 12 log

1+cosω1−cosω .

Proof. A real number t belongs to J(T, θ) iff

e2t sin2 θ + e−2t cos2 θ < e−2T .

Thus J(T, θ) is empty unless

∆ := e−4T − sin2 2θ > 0,which gives the first assertion of the proposition.

When this condition holds, we write sin 2θ = e−2T sinω with cosω > 0. One has∆

12 = e−2T cosω, which implies the second assertion of the proposition through the fol-

lowing elementary calculation.


By performing the change of variables x = e2t, we see that t ∈ J(T, θ) if and only ifx2 sin2 θ − e−2Tx+ cos2 θ < 0, that is, t ∈ J(T, θ) if and only if x = e2t belongs to theopen interval (x−, x+) between the two roots

x± =exp(−2T )±

√∆

2 sin2 θ=

exp(−2T )(1± cosω)2 sin2 θ

In order words, J(T, θ) = (t−, t+) where x± = e2t± . In particular, |J(T, θ)| = t+−t− =12 log

1+cosω1−cosω . �

For later use, we note that

Lemma 3.5. ∫ π2

0

log1 + cosω

1− cosωcosω dω = π.

Proof. The change of variables u = tan ω2 transforms the given integral into 4∫ 1

0log u−1 1−u

2

(1+u2)2 du.

We then have, as∫ 1

0un log u−1 du = (n+ 1)−2 for n > 0∫ 1

0

log u−11− u2

(1 + u2)2du =

∫ 10

log u−1∑n>0

(−1)n(2n+ 1)u2n du

=∑n>0

(−1)n

2n+ 1

=π

4�

Also for later use, let us observe that

Lemma 3.6. Given ω0 > 0, there exists a constant K = K(ω0) > 0 such that ifexp(−2T ) sinω0 < | sin 2θ| < exp(−2T )

for some T > 0 and θ, then‖gtRθe2‖ 6 K exp(−t)

for all t ∈ J(T, θ).

Proof. Note that

‖gtRθe2‖2 = e−2t(cos2 θ + e4t sin2 θ) 6 e−2t(1 + e4t sin2 θ)In particular, from the definition of J(T, θ), we have that e2t sin2 θ 6 ‖gtRθe2‖2 <

exp(−2T ) for t ∈ J(T, θ). By combining these two estimates, we deduce that‖gtRθe2‖2 6 e−2t(1 + exp(−2T )e2t)

On the other hand, by the elementary calculation in the end of the proof of Proposition3.4, we know that, by writing sin 2θ = exp(−2T ) sinω,

e2t 6 x+ := exp(−2T )1 + cosω

2 sin2 θ

for every t ∈ J(T, θ).By hypothesis, exp(−2T ) sinω0 < | sin 2θ| < exp(−2T ) (i.e., ω0 < |ω| < π/2), so

that we conclude from the previous inequality that

exp(−2T )e2t 6 exp(−4T ) 1sin2 θ

64

sin2 ω0


By plugging this into our estimate of ‖gtRθe2‖2 above, we deduce that

‖gtRθe2‖2 6 e−2t(

1 +4

sin2 ω0

):= e−2tK(ω0),

and thus the proof of the lemma is complete. �

4. CONSTRUCTION OF A MEASURE RELATED TO m

In the next three sections, the setting is as indicated in subsection 1.3 and subsection1.5: C is a SL(2,R)-invariant manifold contained in moduli space, and m is a SL(2,R)-invariant probability measure supported on C.

Until further notice, we fix some number ρ > 0, small enough so that the m-measure of{M ∈ C : sys(M) > ρ} is positive.

Let X be the level set {M ∈ C : sys(M) = ρ}. Let X∗0 be the subset of Xformed of surfaces M for which all non-vertical saddle-connections have length > ρ. LetX∗ :=

⋃θ Rθ(X

∗0 ). Observe that in this union, one has Rθ(X

∗0 ) = Rθ+π(X

∗0 ) but also

Rθ0(X∗0 ) ∩Rθ1(X∗0 ) = ∅ for −π2 < θ0 < θ1 6

π2 . Thus we have

X∗ =⊔

−π2


Then the map

Ψ0(u,M) := nu.M

is an smooth diffeomorphism from (−u0, u0)× Σ onto a subset U of X∗0 and the map

Ψ(t, θ, u,M) := gtRθnu.M

is an smooth diffeomorphism from

{(t, θ, u,M) ∈ R× (−π4,π

4)× (−u0, u0)× Σ, 0 < t < log cot |θ|}

onto an open subset V of C. Moreover, one can choose a locally finite covering of X∗0 bysuch sets U . The measure m0 will be defined by its restriction to such sets.

If the m-measure of V is 0, the set U will be disjoint from the support of m0.

Assume that m(V ) > 0. As m is SL(2,R)-invariant, the measure Ψ∗(m|V ) can bewritten, using Proposition 2.6 (with G = SL(2,R)) and Proposition 3.1, as

dt× cos 2θ dθ × du× ν

for some finite measure ν on Σ (cf. Remark 2.8: the open subset Ux of Proposition 2.6 ishere independent of x).

One thus defines

m0|U = (Ψ0)∗(du× ν).

Then m0 satisfies the required conditions. �

In the next corollary, J(T, θ) is the interval that has been defined in Proposition 3.4.

Corollary 4.2. Let B be a Borel subset of X∗0 , ω0 ∈ (0, π2 ], T > 0. Let Y (T, ω0, B) bethe set of surfaces M ′ = gtRθM in Y ∗ such that M ∈ B, | sin 2θ| < exp(−2T ) sinω0and t ∈ J(T, θ). One has

m(Y (T, ω0, B)) =1

4exp(−2T )m0(B)

∫ ω0−ω0

log1 + cosω

1− cosωcosω dω

Proof. Write sin 2θ = e−2T sinω as in Proposition 3.4 , so that the condition | sin 2θ| <exp(−2T ) sinω0 is equivalent to |ω| < ω0. As the length of J(T, θ) is

1

2log

1 + cosω

1− cosω

we obtain from Proposition 4.1

m(Y (T, ω0, B)) = m0(B)

∫| sin 2θ|


5. MEASURE OF THE SLICE {ρ > sys(M) > ρ exp(−τ)}

For ρ > 0, we denote by F (ρ) the m-measure of the set {sys(M) 6 ρ}. This is anon-decreasing function of ρ.

For any ρ > 0, the level set {sys(M) = ρ} has m-measure 0, as may be seen for in-stance by an elementary application of Lemma 1.4. It follows that the function F is contin-uous. We will prove in this section (see Corollary 5.4) that, for any ρ such that F (ρ) < 1,the function F has at ρ a positive left-derivative F ′(ρ) which is equal to πρ−1m0(X∗0 ),where X∗0 , m0 are as Proposition 4.1.

In this section, ρ is as above a positive number such that F (ρ) < 1.

5.1. The regular part of the slice. For M ∈ X∗ and t > 0, we defineΦt(M) := RθgtR−θ(M), when M ∈ Rθ(X∗0 ).

Observe that the two possible choices for θ give the same result as Rπ is the non trivialelement of the center of SL(2,R).

The systole of Φt(M) is equal to ρ exp(−t). For M ∈ Rθ(X∗0 ), all saddle connectionsof Φt(M) with minimal length have the same angle θ with the vertical direction. It followsthat the Φt are injective, and that Φt(X∗) ∩ Φt′(X∗) = ∅ for t 6= t′.

Definition 5.1. For τ > 0, the set⊔

06t6τ Φt(X∗) is called the regular part of the slice

S(τ) := {ρ > sys(M) > ρ exp(−τ)}.Its complement (in S(τ)) is called the singular part of the slice S(τ).

Next, we define, for a Borel subset B of X∗ and τ > 0

m̃τ (B) :=2

1− exp(−2τ)m(

⊔06t6τ

Φt(B)).

Proposition 5.2. The measure m̃τ is independent of τ , and is equal to the product dθ×m0.

Proof. As the measure m̃τ on X∗ is invariant under the action on X∗ of the group ofrotations, it follows from Proposition 2.6 that the measure m̃τ can be written as dθ ×mτ ,for some finite measure mτ on X∗0 . We will show that mτ = m0.

Let Σ and u0 be as in the proof of Proposition 4.1. For (u,M) ∈ (−u0, u0)×Σ, any θand any t > 0, we have

Φt(Rθnu.M) = Rθgtnu.M.

Observe that for t > 0, θ ∈ (−π4 ,π4 ), the element Rθgt belongs to the set W of subsec-

tion 3.1, so that we can write, according to Proposition 3.1

Rθgt = gT (t,θ)RΘ(t,θ)nU(t,θ)

for some smooth functions T,Θ, U . For θ close to 0, we have

T (t, θ) = t+O(θ), U(t, θ) = O(θ), Θ(t, θ) = e−2tθ +O(θ2).

We will use this local information on T (t, θ), Θ(t, θ) and U(t, θ) combined with ourexpression for the measure m|Y ∗ in gTRΘnU -coordinates (cf. Proposition 3.1 and theproof of Proposition 4.1) to compute m̃τ as follows.

LetB0 = (u1, u2)×B be an elementary Borel subset of (−u0, u0)×Σ ⊂ X∗0 , whereBis a Borel subset of Σ. For small θ > 0, we have, in view of the formula for Haar measurein Proposition 3.1


m̃τ ([0, θ]×B0) =2

1− exp(−2τ)

∫B

∫ u2u1

∫ τ0

e−2t θ dt du dν +O(θ2)

= m0(B0) θ +O(θ2).

This proves that mτ = m0. �

5.2. Estimate for the singular part of the slice. Let Z(τ) be the subset of S(τ) consist-ing of surfaces M having a saddle-connection of length 6 ρ exp τ which is not parallel toa minimizing one. Clearly the singular part of the slice S(τ) is contained in Z(τ).

Proposition 5.3. For small τ > 0 , the m-measure of Z(τ) (hence also the m-measure ofthe singular part of the slice S(τ)) is o(τ).

From Propositions 5.2 and 5.3, we get the

Corollary 5.4. One has

limτ→0

1

τm({ρ > sys(M) > ρ exp(−τ)}) = πm0(X∗0 ).

Proof of Proposition 5.3. For M ∈ S(τ), let us denote by θ̂(M) the smallest non-zeroangle between two saddle-connections with length 6 3ρ (If all connections with length6 3ρ are parallel, we define θ̂(M) = π2 ). The required estimate for the measure of Z(τ)follows from two lemmas:

Lemma 5.5. For any η > 0, there exists θ̂0 such that, for any τ > 0 small enough, onehas

m({M ∈ S(τ) : θ̂(M) < θ̂0}) < ητ.

Lemma 5.6. For any θ̂0, one has

m({M ∈ Z(τ) : θ̂(M) > θ̂0}) = O(τ32 ),

where the implied constant depends on θ̂0,ρ and g.

Proof of Lemma 5.5. Denote by S1(τ) the subset of S(τ) consisting of translation surfacesfor which there exists a length-minimizing saddle-connection whose direction form anangle 6 π6 with the vertical direction.

It follows from Lemma 1.4 (with G = SO(2,R) equipped with the Haar measure) thatfor any SO(2,R)-invariant subset S ⊂ S(τ), we have

m(S) 6 3 m(S1(τ) ∩ S).We will apply this relation with

S = {M ∈ S(τ), θ̂(M) < θ̂0},

for some appropriate θ̂0.For M ∈ S1(τ), and any positive integer j such that

exp(3jτ) < cotπ

6=√

3

the systole of g3jτ .M is < ρ exp(−τ). Therefore the images of S1(τ) under the elementsg3jτ , for 0 6 j < log 36τ − 1, are disjoint. Observe also that for such M , j, the surface


M ′ := g3jτ .M has a systole in (ρ2 ,√

3ρ) and has two saddle-connections of length at most3√

3ρ with a non-zero angle 6 Aθ̂(M), for some absolute constant A > 1.

Let η > 0. Let θ̂1 > 0 be small enough in order that the m-measure of the set of sur-faces M ′ with systole in (ρ2 ,

√3ρ), having two non-parallel saddle-connections of length

6 3√

3ρ and angle < θ̂1, is < η18 . Choosing θ̂0 := A−1θ̂1, as the number of values of j

with 0 6 j < log 36τ − 1 is >16τ for τ small enough, we obtain

m({M ∈ S(τ), θ̂(M) < θ̂0} ∩ S1) < 6τ.η

18,

m({M ∈ S(τ), θ̂(M) < θ̂0}) < ητ.�

Proof of Lemma 5.6. We will apply Lemma 1.4 with

B := {M ∈ Z(τ), θ̂(M) > θ̂0},G = SL(2,R), and ν the normalized restriction of a Haar measure to

K := {γ ∈ SL(2,R), ||γ|| 6 2},where || || is the Euclidean operator norm. We therefore have to estimate the relativemeasure in K of the sets {γ ∈ K, γ.M ∈ B}.

If K.M does not intersect B, this measure is 0.Let M ∈ C, γ ∈ Ksuch that γ.M ∈ B . As ||γ|| = ||γ−1|| 6 2, we have

1

2ρ exp(−τ) 6 sys (M) 6 2ρ.

As B ⊂ Z(τ), there exist non colinear holonomy vectors v, v′ of saddle connections ofM such that

ρ exp(−τ) 6 ||γ.v|| 6 ρ exp(τ), ρ exp(−τ) 6 ||γ.v′|| 6 ρ exp(τ).This means that γ belongs to the set E

(ρ−1v, ρ−1v′, τ

)of Proposition 3.3.

As ||γ|| = ||γ−1|| 6 2, we must have (for small τ ) ||v|| 6 3ρ, ||v′|| 6 3ρ.Moreover, the angle between the directions of v, v′ is > A−1θ̂0 for some appropriate

absolute constant A > 1: otherwise, we would have θ̂(γ.M) < θ̂0, in contradiction toγ.M ∈ B. This imply that ||ρ−1(v±v′)|| is bounded from below by a constant c dependingonly on θ̂0.

Let v1, . . . , vN be the holonomy vectors of saddle-connections of M of lengths 6 3ρ.We have shown that

{γ ∈ K : γ.M ∈ B} ⊂⋃E(ρ−1vi, ρ

−1vj , τ)

where the union is taken over indices i, j such that ||ρ−1(vi ± vj)|| > c. In view ofProposition 3.3, we obtain

ν({γ ∈ K : γ.M ∈ B})ν(K)

6 N2 ·Oθ̂0(τ3/2)

By Lemma 1.5 in Subsection 1.6, the integer N has an upper bound depending only onρ and g.

The statement in Lemma 5.6 follows from this estimate, plugged into Lemma 1.4. �

This completes the proof of Proposition 5.3. �


6. PROOF OF THEOREM 1.3

We recall that F (ρ) stands for the m-measure of the set {sys(M) 6 ρ}. It is a continu-ous non-decreasing function of ρ > 0. By Corollary 5.4, at any ρ > 0 such that F (ρ) < 1,the function F has a positive left-derivative F ′(ρ) which is equal to πρ−1m0(X∗0 ) (whereX∗0 , m0 are as in Proposition 4.1) .

Proposition 6.1. For any ρ > 0 with F (ρ) < 1, and any T > 0, we have

F (ρ exp(−T )) > 12

exp(−2T )ρF ′(ρ).

Proof. We apply Corollary 4.2 with B = X∗0 , and ω0 =π2 . From this corollary and

Lemma 3.5, we obtain

m(Y (T,π

2, X∗0 )) =

1

4exp(−2T )m0(X∗0 )

∫ π2

−π2log

1 + cosω

1− cosωcosω dω

=π

2exp(−2T )m0(X∗0 )

=1

2exp(−2T )ρF ′(ρ).

To get the estimate of the proposition, it is now sufficient to recall that, by Proposition3.4, any surface in Y (T, π2 , X

∗0 ) has a systole 6 ρ exp(−T ). �

Corollary 6.2. The function F is absolutely continuous, and its left-derivative F ′(ρ) sat-isfies

F ′(ρ) = O(ρ).

Moreover, defining

c(m) :=1

2sup

F (ρ) C. Given C ′ > C, let

I(C ′) := {ρ ∈ [ρ0, ρ1] : |f(ρ1)− f(ρ)| 6 C ′(ρ1 − ρ)}.

Note that ρ1 ∈ I(C ′) (and hence I(C ′) is not empty), I(C ′) is closed (because f iscontinuous) and I(C ′) is “open to the left” in the sense that, for each ρ∗ ∈ I(C ′), ρ0 <ρ∗ 6 ρ1, there exists δ = δ(ρ∗) > 0 such that [ρ∗ − δ, ρ∗] ⊂ I(C ′) (because of the


bound |f ′(ρ∗)| 6 C < C ′ on the left-derivative). From these properties, we deduce thatI(C ′) = [ρ0, ρ1] and therefore f(ρ1)− f(ρ0)| 6 C(ρ1 − ρ0).

Applying this with f = F , we conclude that F is absolutely continuous.Defining c(m) as in the corollary, we get from Proposition 6.1 that

lim infρ→0

F (ρ)

ρ2> c(m).

On the other hand, since F is absolutely continuous, it has a derivative at almost everypoint and it is the integral of its almost everywhere derivative. In particular, in our setting,it follows that F is the integral of its left-derivative F ′, so that

lim supρ→0

F (ρ)

ρ26 c(m).

This proves that limρ→0 ρ−2F (ρ) = c(m). The last assertion of the corollary is thenobvious. �

Proof of Theorem 1.3. We have to prove that the m-measure of the set of M ∈ C whichhave at least two non-parallel saddle-connections of length 6 ρ is o(ρ2) when ρ is small.Taking into account Remark 1.1, it is sufficient to prove that, for anyA > 1, the set C(A, ρ)formed by the pointsM ∈ C with sys(M) 6 ρ having another saddle-connection of length6 A · sys(M), non parallel to the minimizing one, has m-measure o(ρ2) when ρ is small.

Fix some A > 1, and some η > 0.We will prove that

(6.1) m(C(A, ρ)) < ηρ2,

when ρ is sufficiently small.

We first choose ρ0 sufficiently small to satisfy F (ρ0) < 1 and

(6.2) ρ−10 F′(ρ0) > 2c(m)−

η

2;

(6.3) ρ−2F (ρ) < (c(m) +η

4) for 0 < ρ < ρ0.

We use ρ0 as the level of the systole function at which are defined X∗0 , m0 in Section 4.

Let ω0 = ω0(η) > 0 be small enough to have

(6.4) m0(X∗0 )∫ ω0−ω0

log1 + cosω

1− cosωcosω dω < ηρ20.

Next, consider T > 0 and θ such that exp(−2T ) sinω0 < | sin 2θ| < exp(−2T ). ByLemma 3.6, there exists K = K(ω0), independent of T , such that, for all t ∈ J(T, θ), thenorm of the image of the vertical basis vector e2 under gtRθ is 6 K exp(−t).

Thus, if v is any vector with ||v|| > AK, we will have, for such θ and t

(6.5) ||gtRθv|| > AK exp(−t) > A||gtRθe2||.


LetM ∈ X∗0 . Denote by θ̄(M) the minimal angle between a length-minimizing saddle-connection and another non-parallel saddle connection of length6 AKρ0. If no such shortnon-parallel saddle connection exists, set θ̄(M) = π2 .

As θ̄ is everywhere positive on X∗0 , there exists θ̄0 such that the set B(θ̄0) := {M ∈X∗0 , θ̄(M) < θ̄0} satisfies

(6.6)π

2m0(B(θ̄0)) <

η

4ρ20.

On the other hand, if T is sufficiently large (say T > T0 = T0(A, θ̄0)), one has, for| sin 2θ| < exp(−2T ) , π2 > |θ

′| > θ̄0, t ∈ J(T, θ)

(6.7) ||gtRθ+θ′e2|| > A||gtRθe2||.

In fact, recall that t ∈ J(T, θ) if and only if ‖gtRθe2‖2 < exp(−2T ) and J(T, θ) 6= ∅ ifand only if | sin 2θ| < exp(−2T ), cf. Proposition 3.4. In particular, for T large enough(depending on θ̄0), we have that |θ| 6 θ̄0/2. It follows that, for any |θ′| > θ̄0, one has

‖gtRθ+θ′e2‖2 = e2t sin2(θ + θ′) + e−2t cos2(θ + θ′) > e2t sin2(θ̄0/2)

Now, we observe that the proof of Proposition 3.4 also gives that

e2t > exp(−2T )1− cosω2 sin2 θ

for any t ∈ J(T, θ), where sin 2θ = exp(−2T ) sinω. From this discussion, we obtain

‖gtRθ+θ′e2‖2 > sin2(θ̄0/2) exp(−2T )1− cosω2 sin2 θ

> sin2(θ̄0/2)1− cosω2 sin2 θ

‖gtRθe2‖2

for T large enough (depending on θ̄0) and t ∈ J(T, θ). On the other hand, since sin 2θ =exp(−2T ) sinω, if T is large than an absolute constant so that | cos θ| > 1/2, then

2(1− cosω) > 1− cos2 ω = sin2 ω = exp(4T ) sin2 2θ > exp(4T ) sin2 θ,

that is, (1 − cosω)/ sin2 θ > exp(4T )/2. So, by putting these estimates together, weobtain that if T is large enough (depending on θ̄0 and A) and |θ′| > θ̄0, then

‖gtRθ+θ′e2‖2 > sin2(θ̄0/2)(exp(4T )/4)‖gtRθe2‖2 > A2‖gtRθe2‖2,

that is, (6.7) holds.

We claim that (6.1) holds for ρ 6 ρ0 exp(−T0). Indeed, writing ρ = ρ0 exp(−T ) withT > T0:

• The measure of elements in {sys(M) 6 ρ} which are not in Y (T, π2 , X∗0 ) is

< η2ρ2. Indeed, from (6.2), (6.3), Corollary 4.2 and Proposition 6.1, we have

F (ρ) <1

2(c(m) +

η

2)ρ2

and

m(Y (T,π

2, X∗0 )) =

1

2exp(−2T )ρ0F ′(ρ0) >

1

2ρ2(c(m)− η

2).

• For M ∈ Y (T, π2 , X∗0 ), write M = gtRθM0, with M0 ∈ X∗0 , | sin 2θ| <

exp(−2T ), t ∈ J(T, θ) . From (6.4) and Corollary 4.2, we get

m(Y (T, ω0, X∗0 )) <

1

4ηρ2.


Similarly, from (6.6) and Corollary 4.2, we get

m(Y (T,π

2, B(θ̄0))) <

1

4ηρ2.

• Assume that M = gtRθM0, with θ̄(M0) > θ̄0, | sin 2θ| > exp(−2T ) sinω0.Assume also that T > T0. Then the point M does not belong to C(A, ρ). Thisfollows from (6.5) for the saddle connections of M0 of length > AKρ0, and from(6.7) for the saddle connections of M0 of length 6 AKρ0.

This concludes the proof of (6.1) and also of the theorem. �

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CNRS UMR 7586, INSTITUT DE MATHÉMATIQUES DE JUSSIEU - PARIS RIVE GAUCHE, BÂTIMENTSOPHIE GERMAIN, CASE 7012, 75205 PARIS CEDEX 13, FRANCE & IMPA, ESTRADA DONA CASTORINA110, 22460-320, RIO DE JANEIRO, BRAZIL

E-mail address: [email protected]

UNIVERSITÉ PARIS 13, SORBONNE PARIS CITÉ, CNRS (UMR 7539), F-93430, VILLETANEUSE, FRANCE.E-mail address: [email protected]

COLLÈGE DE FRANCE (PSL), 3, RUE D’ULM, 75005 PARIS, FRANCEE-mail address: [email protected]

1. Introduction1.1. Regular SL(2,R)-invariant probability measures on Hg1.2. The result1.3. Reduction to a particular case1.4. Scheme of the proof 1.5. Notations1.6. Two basic factsAcknowledgements

2. Conditional measures2.1. The general setting2.2. A special setting

3. Preliminaries on SL(2,R)3.1. The decomposition gt R nu3.2. Euclidean norms along a SL(2,R)-orbit3.3. On the action of the diagonal subgroup

4. Construction of a measure related to m5. Measure of the slice {sys(M) exp(-)}5.1. The regular part of the slice5.2. Estimate for the singular part of the slice

6. Proof of Theorem 1.3References

Documents

arXiv:1302.4091v2 [math.DS] 29 Jun 2013w3.impa.br/~cmateus/files/AMY1.pdf · 2 A. AVILA, C. MATHEUS AND J.-C. YOCCOZ relationship with the moduli spaces H gof normalized (unit area)